"sign, the equation would in some cases describe only half a circle. In the complex plane, a circle with a centre at ""c"" and radius ""r"" has the equation: In parametric form, this can be written: The slightly generalised equation for real ""p"", ""q"" and complex ""g"" is sometimes called a generalised circle. This becomes the above equation for a circle with formula_21, since formula_22. Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a line. The tangent line through a point ""P"" on the circle is perpendicular to the diameter passing through"
"Generalised circle A generalized circle, also referred to as a ""cline"" or ""circline"", is a straight line or a circle. The concept is mainly used in inversive geometry, because straight lines and circles have very similar properties in that geometry and are best treated together. Inversive plane geometry is formulated on the plane extended by one point at infinity. A straight line is then thought of as one of the circles that passes through the asymptotic point at infinity. The fundamental transformations in inversive geometry, the ""inversions"", have the property that they map generalized circles to generalized circles. Möbius transformations,"
"Trochoid A trochoid (from the Greek word for wheel, ""trochos"") is the curve described by a fixed point on a circle as it rolls along a straight line. The cycloid is a notable member of the trochoid family. The word ""trochoid"" was coined by Gilles de Roberval. As a circle of radius ""a"" rolls without slipping along a line ""L,"" the center ""C"" moves parallel to ""L,"" and every other point ""P"" in the rotating plane rigidly attached to the circle traces the curve called the trochoid. Let ""CP = b"". Parametric equations of the trochoid for which ""L"" is"
"Circle A circle is a simple closed shape. It is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant. The distance between any of the points and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. A circle is a simple closed curve that divides the plane into two regions: an interior"
"Central line (geometry) In geometry central lines are certain special straight lines associated with a plane triangle and lying in the plane of the triangle. The special property that distinguishes a straight line as a central line is manifested via the equation of the line in trilinear coordinates. This special property is related to the concept of triangle center also. The concept of a central line was introduced by Clark Kimberling in a paper published in 1994. Let ""ABC"" be a plane triangle and let ( ""x"" : ""y"" : ""z"" ) be the trilinear coordinates of an arbitrary point"
"It ""is"" possible to do them with straightedge alone given one circle and its center. Straightedge A straightedge or straight edge is a tool used for drawing straight lines, or checking their straightness. If it has equally spaced markings along its length, it is usually called a ruler. Straightedges are used in the automotive service and machining industry to check the flatness of machined mating surfaces. True straightness can in some cases be checked by using a laser line level as an optical straightedge: it can illuminate an accurately straight line on a flat surface such as the edge of"
"Chord (geometry) A chord of a circle is a straight line segment whose endpoints both lie on the circle. A secant line, or just ""secant"", is the infinite line extension of a chord. More generally, a chord is a line segment joining two points on any curve, for instance an ellipse. A chord that passes through a circle's center point is the circle's diameter. Every diameter is a chord, but not every chord is a diameter. The word ""chord"" is from the Latin ""chorda"" meaning ""bowstring"". Among properties of chords of a circle are the following: The midpoints of a"
"line cannot be drawn without a straightedge. A line is considered to be given by any two points, as any two points define a line uniquely, and a unique line can be defined by any two points on it. This can be done with compass alone quite naturally. There is nothing to prove. This construction can be done directly with a compass provided the centers and radii of the two circles are known. Due to the compass-only construction of the center of a circle (given below), it can always be assumed that any circle is described by its center and"
"Centered trochoid In geometry, a centered trochoid is the roulette formed by a circle rolling along another circle. That is, it is the path traced by a point attached to a circle as the circle rolls without slipping along a fixed circle. The term encompasses both epitrochoid and hypotrochoid. The center of this curve is defined to be the center of the fixed circle. Alternatively, a centered trochoid can be defined as the path traced by the sum of two vectors, each moving at a uniform speed in a circle. Specifically, a centered trochoid is a curve that can be"
"because formula_5 Apothem The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. The word ""apothem"" can also refer to the length of that line segment. Regular polygons are the only polygons that have apothems. Because of this, all the apothems in a polygon will be congruent. For a regular pyramid, which is a pyramid whose base is a regular polygon, the apothem is the"
"fact the only exception) to the dual generation theorem stated above. This degenerate case, in which the curve is a straight-line segment, underlies the Tusi-couple. Centered trochoid In geometry, a centered trochoid is the roulette formed by a circle rolling along another circle. That is, it is the path traced by a point attached to a circle as the circle rolls without slipping along a fixed circle. The term encompasses both epitrochoid and hypotrochoid. The center of this curve is defined to be the center of the fixed circle. Alternatively, a centered trochoid can be defined as the path traced"
"Hypercycle (geometry) In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line (its axis). Given a straight line L and a point P not on L, one can construct a hypercycle by taking all points Q on the same side of L as P, with perpendicular distance to L equal to that of P. The line L is called the ""axis"", ""center"", or ""base line"" of the hypercycle. The lines perpendicular to the ""axis"", which is also perpendicular to the hypercycle are called the ""normals"" of"
"called a hypercycle. Another special curve is the horocycle, a curve whose normal radii (perpendicular lines) are all limiting parallel to each other (all converge asymptotically in one direction to the same ideal point, the centre of the horocycle ). Through every pair of points there are two horocycles. The centres of the horocycles are the ideal points of the perpendicular bisector of the line-segment between them. Given any three distinct points, they all lie on either a line, hypercycle, horocycle, or circle. The length of the line-segment is the shortest length between two points. The arc-length of a hypercycle"
"the same points, but at right angles. Hypercycle (geometry) In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line (its axis). Given a straight line L and a point P not on L, one can construct a hypercycle by taking all points Q on the same side of L as P, with perpendicular distance to L equal to that of P. The line L is called the ""axis"", ""center"", or ""base line"" of the hypercycle. The lines perpendicular to the ""axis"", which is also perpendicular to"
"Osculating circle In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ""p"" on the curve has been traditionally defined as the circle passing through ""p"" and a pair of additional points on the curve infinitesimally close to ""p"". Its center lies on the inner normal line, and its curvature is the same as that of the given curve at that point. This circle, which is the one among all tangent circles at the given point that approaches the curve most tightly, was named ""circulus osculans"" (Latin for ""kissing circle"") by Leibniz."
"red circle with an angular line crossing it. Either overtaking is prohibited for all vehicles or certain kinds of vehicles only (e.g. lorries, motorcycles, etc...). In the USA, this is usually phrased as ""no passing zone"" and indicated by a rectangular, black-on-white sign on the right side of the road that says ""DO NOT PASS"" and/or by a solid yellow line painted on the roadway marking the left limit of traffic (centerline), and sometimes supplemented by a yellow (no passing zone) sign on the opposite side of the road (where it can be seen by a driver who is attempting"
"Circular segment In geometry, a circular segment (symbol: ⌓) is a region of a circle which is ""cut off"" from the rest of the circle by a secant or a chord. More formally, a circular segment is a region of two-dimensional space that is bounded by an arc (of less than 180°) of a circle and by the chord connecting the endpoints of the arc. Let ""R"" be the radius of the circle, ""θ"" the central angle in radians, α is the central angle in degrees, ""c"" the chord length, ""s"" the arc length, ""h"" the sagitta (height) of the"
"Radical axis The radical axis (or power line) of two circles is the locus of points at which tangents drawn to both circles have the same length. The radical axis is always a straight line and always perpendicular to the line connecting the centers of the circles, albeit closer to the circumference of the larger circle. If the circles intersect, the radical axis is the line passing through the intersection points; similarly, if the circles are tangent, the radical axis is simply the common tangent. For any point P on the radical axis, there is a unique circle centered on"
"notes on the study of curvature, see For application to maneuvering vehicles see Osculating circle In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ""p"" on the curve has been traditionally defined as the circle passing through ""p"" and a pair of additional points on the curve infinitesimally close to ""p"". Its center lies on the inner normal line, and its curvature is the same as that of the given curve at that point. This circle, which is the one among all tangent circles at the given point that approaches the"
"Diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere. In more modern usage, the length of a diameter is also called the diameter. In this sense one speaks of ""the"" diameter rather than ""a"" diameter (which refers to the line itself), because all diameters of a circle or sphere have the same length, this being twice the radius"
"Squircle A squircle is a mathematical shape intermediate between a square and a circle. It is a special case of a superellipse. The word ""squircle"" is a portmanteau of the words ""square"" and ""circle"". In a Cartesian coordinate system, the squircle centred on the point (""a"", ""b"") with axes parallel to the coordinate axes is described by the equation: where ""r"" is the minor radius of the squircle (cf. equation of a circle). The case that is centred on the origin (that is, with ""a"" = ""b"" = 0) is called Lamé's special quartic. When ""a"" = ""b"" = 0"
"Cardioid A cardioid (from the Greek καρδία ""heart"") is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion. The name was coined by de Castillon in 1741 but had been the subject of study decades beforehand. Named for its heart-like form, it is shaped more like the outline of the"
"Circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. A polygon that has a circumscribed circle is called a cyclic polygon (sometimes a concyclic polygon, because the vertices are concyclic). All regular simple polygons, all isosceles trapezoids, all triangles and all rectangles are cyclic. A related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it. Not every"
"is sometimes called a ""double cone"". Either half of a double cone on one side of the apex is called a ""nappe"". The axis of a cone is the straight line (if any), passing through the apex, about which the base (and the whole cone) has a circular symmetry. In common usage in elementary geometry, cones are assumed to be right circular, where ""circular"" means that the base is a circle and ""right"" means that the axis passes through the centre of the base at right angles to its plane. If the cone is right circular the intersection of a"
"a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at its vertices. As mentioned above, every triangle has a unique circumcircle, a circle passing through all three vertices, whose center is the intersection of the perpendicular bisectors of the triangle's sides. Further, every triangle has a unique Steiner circumellipse, which passes through the triangle's vertices and has its center at the triangle's centroid. Of all ellipses going through the triangle's vertices, it has the smallest area. The Kiepert hyperbola is the unique conic which passes through the triangle's three vertices, its"
"Cross A cross is a geometrical figure consisting of two intersecting lines or bars, usually perpendicular to each other. The lines usually run vertically and horizontally. A cross of oblique lines, in the shape of the Latin letter X, is also termed a saltire in heraldic terminology. The word ""cross"" is recorded in 10th-century Old English as ""cros"", exclusively for the instrument of Christ's crucifixion, replacing the native Old English word ""rood"". The word's history is complicated; it appears to have entered English from Old Irish, possibly via Old Norse, ultimately from the Latin ""crux"" (or its accusative ""crucem"" and"
"Cycloid A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve. The cycloid, with the cusps pointing upward, is the curve of fastest descent under constant gravity, and is also the form of a curve for which the period of an object in descent on the curve does not depend on the object's starting position. The cycloid has been"
"has the Euler line as its directrix. Euler line In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle. The concept of a triangle's Euler line extends to the Euler line of other shapes, such as the quadrilateral and the tetrahedron. Euler showed in 1765 that in any triangle, the"
"Line (geometry) The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth. Lines are an idealization of such objects. Until the 17th century, lines were defined as the ""[…] first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width. […] The straight line is that which is equally extended between its"
"path similar to a figure-eight or infinity symbol. An Around is a trick in which the pen rotates around one or more body parts and is named after the body part(s) the pen rotates around. For example, in a Middle Index Around, the pen goes around the middle and index fingers. The Thumb Around, as a fundamental, is usually the first Around a pen spinner learns. When an Around Reverse is executed fingerlessly with a turn of the hand or wrist, it is known as a Back Around, or simply ""Bak."" Baks, while technically a type of Around, are often"
"Circular sector A circular sector or circle sector (symbol: ⌔), is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. In the diagram, θ is the central angle in radians, formula_1 the radius of the circle, and formula_2 is the arc length of the minor sector. A sector with the central angle of 180° is called a half-disk and is bounded by a diameter and a semicircle. Sectors with other central angles are sometimes given special names, these include quadrants"
"then circumscribe a square. Again circumscribe a circle, then circumscribe a regular 5-gon, and so on. The radii of the circumscribed circles converge to the so-called ""polygon circumscribing constant"" . The reciprocal of this constant is the Kepler–Bouwkamp constant. Circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. A polygon that has a circumscribed circle is called a cyclic polygon (sometimes a concyclic polygon, because the vertices are concyclic)."
"a straight line, such that the acceleration is directed towards the center of the motion and also proportional to the displacement from that point. The motion of a particle moving along a straight line with an acceleration whose direction is always towards a fixed point on the line and whose magnitude is proportional to the distance from the variable point is called simple harmonic motion [SHM]. In the diagram, a simple harmonic oscillator, consisting of a weight attached to one end of a spring, is shown. The other end of the spring is connected to a rigid support such as"